Subgroups — C₁₃×#12929 #15699268

Order 3328.

458 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#365	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2112	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1810	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1383	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#106	—	2	1	no	—
12	C₂×C₁₃×#106	—	2	4	no	—
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₁₃×#407	—	2	1	no	—
15	C₁₃×#365	—	2	4	no	—
16	C₁₃×C₂²⋊C₁₆	C₄	1	4	yes	complemented
17	C₁₃×#341	—	2	1	no	—
18	C₂³×C₄×C₁₃	—	4	1	no	—
19	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	1	no	—
21	C₂×C₁₃×M₁₆	—	2	1	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
32	C₁₃×M₃₂	—	2	2	no	—
33	C₁₃×M₃₂	—	2	2	no	—
34	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
35	C₁₃×#108	—	2	2	no	—
36	C₁₃×#108	—	2	2	no	—
37	C₁₃×#106	—	2	4	no	—
38	C₁₃×#106	—	2	2	no	—
39	C₁₃×#106	D₄	1	4	yes	—
40	C₁₃×C₂²⋊C₈	—	4	2	no	—
41	#12929	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	4	4	no	—
48	C₂²×C₄×C₁₃	—	4	2	no	—
49	C₂²×C₄×C₁₃	—	8	4	no	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₁₃×M₁₆	—	4	1	no	—
55	C₁₃×M₁₆	—	4	1	no	—
56	C₁₃×M₁₆	—	4	1	no	—
57	C₁₃×M₁₆	—	4	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	—	4	1	no	—
63	C₂×C₈×C₁₃	—	4	1	no	—
64	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₁₃	#41	1	2	yes	—
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	4	4	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₁₃×C₁₆	—	2	4	no	—
73	C₂²×#106	C₂×C₁₃	1	1	yes	characteristic
74	C₂×#365	C₂×C₁₃	1	2	yes	complemented
75	#2112	C₂×C₁₃	1	1	yes	characteristic
76	#1810	C₂×C₁₃	1	1	yes	characteristic
77	#1383	C₂×C₁₃	1	2	yes	complemented
78	C₂³×C₁₃	—	2	4	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	C₂³×C₁₃	#106	1	2	yes	—
82	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	4	4	no	—
90	C₂×C₄×C₁₃	—	8	8	no	—
91	C₂×C₄×C₁₃	—	8	4	no	—
92	C₂×C₄×C₁₃	#106	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
99	C₈×C₁₃	—	2	2	no	—
100	C₈×C₁₃	—	8	1	no	—
101	C₈×C₁₃	—	8	1	no	—
102	C₈×C₁₃	C₄×D₄	1	2	yes	—
103	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
104	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
105	C₂²×C₄⋊C₄	—	2	1	no	—
106	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
107	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
108	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
109	C₂×#106	—	2	1	no	—
110	C₂×#106	—	2	4	no	—
111	C₂×#106	C₂²×C₁₃	1	2	yes	—
112	#407	—	2	1	no	—
113	#365	—	2	4	no	—
114	C₂²⋊C₁₆	C₄×C₁₃	1	4	yes	complemented
115	#341	—	2	1	no	—
116	C₂²×C₁₃	—	2	4	no	—
117	C₂²×C₁₃	—	2	4	no	—
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	#365	1	4	yes	complemented
122	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
123	C₂²×C₁₃	#450	1	1	yes	characteristic
124	C₂²×C₁₃	#483	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	8	4	no	—
127	C₄×C₁₃	#385	1	1	yes	characteristic
128	C₄×C₁₃	#428	1	1	yes	characteristic
129	C₄×C₁₃	#448	1	1	yes	characteristic
130	C₄×C₁₃	#481	1	1	yes	characteristic
131	C₂³×C₄	—	4	1	no	—
132	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
133	C₂×M₁₆	—	2	1	no	—
134	C₂×M₁₆	—	2	1	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
138	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
139	C₂²×C₈	Q₈×C₁₃	1	2	yes	—
140	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	2	4	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	4	2	no	—
144	C₂×C₄⋊C₄	—	4	1	no	—
145	M₃₂	—	2	2	no	—
146	M₃₂	—	2	2	no	—
147	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	complemented
148	#108	—	2	2	no	—
149	#108	—	2	2	no	—
150	#106	—	2	4	no	—
151	#106	—	2	2	no	—
152	#106	D₄×C₁₃	1	4	yes	—
153	C₂²⋊C₈	—	4	2	no	—
154	C₂×C₁₃	—	2	1	no	—
155	C₂×C₁₃	—	2	4	no	—
156	C₂×C₁₃	—	2	1	no	—
157	C₂×C₁₃	#1853	1	1	yes	characteristic
158	C₂×C₁₃	C₂×#365	1	2	yes	—
159	C₂⁴	C₁₃×SD₁₆	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	8	4	no	—
167	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
168	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
169	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
171	M₁₆	—	4	1	no	—
172	M₁₆	—	4	1	no	—
173	M₁₆	—	4	1	no	—
174	M₁₆	—	4	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
184	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
185	C₂×C₈	C₁₃×#41	1	2	yes	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₁₆	—	2	4	no	—
190	C₁₃	#12929	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³	—	2	4	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	C₁₃×#106	1	2	yes	—
195	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	8	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	C₁₃×#106	1	2	yes	—
206	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
208	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
209	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
210	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
211	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	8	1	no	—
214	C₈	—	8	1	no	—
215	C₈	C₄×D₄×C₁₃	1	2	yes	—
216	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	C₁₃×#365	1	4	yes	complemented
223	C₂²	C₂×C₁₃×#106	1	1	yes	characteristic
224	C₂²	C₁₃×#450	1	1	yes	characteristic
225	C₂²	C₁₃×#483	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	8	4	no	—
228	C₄	C₁₃×#385	1	1	yes	characteristic
229	C₄	C₁₃×#428	1	1	yes	characteristic
230	C₄	C₁₃×#448	1	1	yes	characteristic
231	C₄	C₁₃×#481	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	4	no	—
234	C₂	—	2	1	no	—
235	C₂	C₁₃×#1853	1	1	yes	characteristic
236	C₂	C₂×C₁₃×#365	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂⁴×C₁₃	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₁₃×#365	C₂	semidirect
5. C₁₃×#1383	C₂	semidirect
16. C₁₃×C₂²⋊C₁₆	C₄	semidirect
34. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
41. #12929	C₁₃	direct (N × H)
74. C₂×#365	C₂×C₁₃	semidirect
77. #1383	C₂×C₁₃	semidirect
114. C₂²⋊C₁₆	C₄×C₁₃	semidirect
121. C₂²×C₁₃	#365	semidirect
147. C₂×C₁₆	C₂×C₄×C₁₃	semidirect
190. C₁₃	#12929	direct (N × H)
222. C₂²	C₁₃×#365	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₃×#106	2
C₂×C₁₃×#365	2
C₁₃×#2112	2
C₁₃×#1810	2
C₁₃×#1383	2
#12929	13

Subgroups — C13×#12929 #15699268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12929 #15699268